Scaling potentials and the billiard case

The potential contours all have the same shape, and the radial
dependence is a -power-law in any direction ( is even
for bounded motion).
This gives the property
everywhere in space,
so that the constant of the motion
can be written
.
This last expression is composed of integrand terms in (D.8),
in the case of isotropic
,
corresponding to dilation.
Thus in this case the integrand is constant [and equal to ,
in order to have zero time-average of
];
there is no diffusive growth and dilation is `special'.
The limit of hard-walled billiards corresponds to
,
in which case the constant of the motion is simply the kinetic term .

To summarize, in dimensions in a scaling potential (of which the
billiard is a special case),
counting the `special' degrees of freedom gives:
for translations (vector
),
for rotations (antisymmetric
part of ), and 1 for dilation (isotropic part of ).
The total is
.

I have strong numerical evidence that dilation is the only new special
deformation which always arises when a hard-walled limit is taken
of a general potential.
Certainly the above arguments are sufficient to exclude simple cases, such
as shear-type deformations.
However I cannot exclude the possibility that a
which is not differentiable everywhere in space allows new
special
functions to arise, which are
not expressible as the Taylor series of (D.6).
Also worthy of study is the general Hamiltonian system
, no longer
restricted to a constant mass tensor
.
This restriction played a key role
in the above arguments.